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Cometric Association Schemes

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Bill Martin

Worcester Polytechnic Institute

USA

Cometric Association Schemes

Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008

- Jason Williford
- Misha Muzychuk
- Edwin van Dam
- Nick LeCompte (WPI student)
- Will Owens (WPI student)
- . . . and I’ve received valuable suggestions from many others.

- Survey the known examples
- Summarize the main results to date
- Explore the structure of imprimitive
Q-polynomial schemes, especially with

3 or 4 classes

- List some open problems, big and small

- To make the next 45 minutes as pleasant as possible

- To make the next 45 minutes as pleasant as possible (for both you and me)

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb
- To get some smart people to work on these interesting problems

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb
- To get some smart people to work on these interesting problems
- To tell you as much as I reasonably can about the subject

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb
- To get some smart people to work on these interesting problems
- To tell you as much as I reasonably can about the subject
- To avoid typesetting math in PowerPoint

E8 Root Lattice

Inner product of two zonal polynomials only depends on distance between the two base points and the single-variable polynomials.

Delsarte (1973):

Concerning cometric association schemes . . .

- I don’t know
- The model I just showed you is my favorite definition so far

Terwilliger (1987):

- Q-polynomial distance-regular graphs (e.g., all those with classical parameters)
- Spherical designs / lattices
- Extremal codes and block designs
- Real mutually unbiased bases
- Sporadic groups (e.g., triality)
- linked systems of designs and geometries

w=3 fibres of size r=2

w=2 fibres of size r=3

A familiar dual pair of association schemes

Another dual pair of complete multipartite schemes

H. Suzuki (1998):

Edwin van Dam (1995)

This is a 4-class Q-antipodal association scheme

A Construction of Wocjan and Beth (2005)

A Construction of Wocjan and Beth (2005)

- 48 vertices, split into three classes of size 16
- Graph G1represents “incidence”, yielding a
square (16,6,2)-design between any two

Q-antipodal classes

- “linked”: the number of common neighbors in the third class of a point chosen from Class One and a point chosen from Class Two depends on only whether or not these are incident (1 and 3, resp.)

- Muzychuk, Williford, WJM introduced the extended Q-bipartite double
- Applied to the subschemes of the Cameron-Seidel scheme, these are 4-class Q-bipartite, Q-antipodal schemes
- So we have the same schemes that Bannai and Bannai found from mutually unbiased bases

Check time available

- 196560 vectors in R24, all of squared length 8
- only 7 possible inner products: ±8, ±4, ±2, 0
- construct one graph for each inner product
- we obtain a 7-class cometric scheme which is Q-bipartite
- Krein array:
- {24, 23, 288/13, 150/7, 104/5, 81/4;
- 1, 24/13, 18/7, 16/5, 15/4, 24 }

Heather Lewis and …?